↑

Properties

Label 26.0.46973965240...7683.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,3^{13}\cdot 13^{48}$
Root discriminant $197.27$
Ramified primes $3, 13$
Class number $7171177$ (GRH)
Class group $[13, 551629]$ (GRH)
Galois group $C_{26}$ (as 26T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![148035889, 4427364461, 130850955287, 55803125274, 154621048686, 33671529801, 106111400109, 15685044765, 42930078217, 1115139675, 11474018296, -454924431, 2237301599, -209210171, 319252037, -35475531, 33363850, -3861598, 2466438, -221065, 118209, -7527, 4004, -130, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 78*x^24 - 130*x^23 + 4004*x^22 - 7527*x^21 + 118209*x^20 - 221065*x^19 + 2466438*x^18 - 3861598*x^17 + 33363850*x^16 - 35475531*x^15 + 319252037*x^14 - 209210171*x^13 + 2237301599*x^12 - 454924431*x^11 + 11474018296*x^10 + 1115139675*x^9 + 42930078217*x^8 + 15685044765*x^7 + 106111400109*x^6 + 33671529801*x^5 + 154621048686*x^4 + 55803125274*x^3 + 130850955287*x^2 + 4427364461*x + 148035889)
 
gp: K = bnfinit(x^26 + 78*x^24 - 130*x^23 + 4004*x^22 - 7527*x^21 + 118209*x^20 - 221065*x^19 + 2466438*x^18 - 3861598*x^17 + 33363850*x^16 - 35475531*x^15 + 319252037*x^14 - 209210171*x^13 + 2237301599*x^12 - 454924431*x^11 + 11474018296*x^10 + 1115139675*x^9 + 42930078217*x^8 + 15685044765*x^7 + 106111400109*x^6 + 33671529801*x^5 + 154621048686*x^4 + 55803125274*x^3 + 130850955287*x^2 + 4427364461*x + 148035889, 1)
 

Normalized defining polynomial

\( x^{26} + 78 x^{24} - 130 x^{23} + 4004 x^{22} - 7527 x^{21} + 118209 x^{20} - 221065 x^{19} + 2466438 x^{18} - 3861598 x^{17} + 33363850 x^{16} - 35475531 x^{15} + 319252037 x^{14} - 209210171 x^{13} + 2237301599 x^{12} - 454924431 x^{11} + 11474018296 x^{10} + 1115139675 x^{9} + 42930078217 x^{8} + 15685044765 x^{7} + 106111400109 x^{6} + 33671529801 x^{5} + 154621048686 x^{4} + 55803125274 x^{3} + 130850955287 x^{2} + 4427364461 x + 148035889 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 13]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-469739652406953148168948870108145354763666849944084892337683=-\,3^{13}\cdot 13^{48}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $197.27$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(507=3\cdot 13^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{507}(1,·)$, $\chi_{507}(131,·)$, $\chi_{507}(196,·)$, $\chi_{507}(326,·)$, $\chi_{507}(391,·)$, $\chi_{507}(14,·)$, $\chi_{507}(79,·)$, $\chi_{507}(209,·)$, $\chi_{507}(274,·)$, $\chi_{507}(404,·)$, $\chi_{507}(469,·)$, $\chi_{507}(92,·)$, $\chi_{507}(157,·)$, $\chi_{507}(287,·)$, $\chi_{507}(352,·)$, $\chi_{507}(482,·)$, $\chi_{507}(40,·)$, $\chi_{507}(170,·)$, $\chi_{507}(235,·)$, $\chi_{507}(365,·)$, $\chi_{507}(430,·)$, $\chi_{507}(53,·)$, $\chi_{507}(118,·)$, $\chi_{507}(248,·)$, $\chi_{507}(313,·)$, $\chi_{507}(443,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{7}{19} a^{14} - \frac{5}{19} a^{13} + \frac{7}{19} a^{11} + \frac{2}{19} a^{10} + \frac{7}{19} a^{9} - \frac{8}{19} a^{8} - \frac{8}{19} a^{7} - \frac{9}{19} a^{6} - \frac{1}{19} a^{5} - \frac{3}{19} a^{4} - \frac{8}{19} a^{3} + \frac{4}{19} a^{2} + \frac{9}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{16} + \frac{3}{19} a^{14} + \frac{3}{19} a^{13} + \frac{7}{19} a^{12} - \frac{6}{19} a^{11} + \frac{2}{19} a^{10} + \frac{3}{19} a^{9} - \frac{7}{19} a^{8} - \frac{8}{19} a^{7} - \frac{7}{19} a^{6} + \frac{9}{19} a^{5} + \frac{9}{19} a^{4} + \frac{5}{19} a^{3} - \frac{1}{19} a^{2} - \frac{9}{19} a + \frac{9}{19}$, $\frac{1}{437} a^{17} - \frac{10}{437} a^{16} + \frac{9}{437} a^{15} + \frac{45}{437} a^{14} + \frac{4}{437} a^{13} - \frac{9}{23} a^{12} - \frac{181}{437} a^{11} + \frac{166}{437} a^{10} + \frac{214}{437} a^{9} + \frac{204}{437} a^{8} + \frac{158}{437} a^{7} - \frac{70}{437} a^{6} + \frac{217}{437} a^{5} - \frac{103}{437} a^{4} - \frac{61}{437} a^{3} + \frac{6}{437} a^{2} - \frac{189}{437} a - \frac{7}{19}$, $\frac{1}{437} a^{18} + \frac{1}{437} a^{16} - \frac{3}{437} a^{15} - \frac{52}{437} a^{14} - \frac{39}{437} a^{13} + \frac{64}{437} a^{12} - \frac{103}{437} a^{11} + \frac{34}{437} a^{10} - \frac{94}{437} a^{9} + \frac{36}{437} a^{8} + \frac{130}{437} a^{7} + \frac{5}{19} a^{6} - \frac{26}{437} a^{5} + \frac{151}{437} a^{4} + \frac{86}{437} a^{3} + \frac{101}{437} a^{2} - \frac{188}{437} a - \frac{1}{19}$, $\frac{1}{437} a^{19} + \frac{7}{437} a^{16} + \frac{8}{437} a^{15} - \frac{130}{437} a^{14} + \frac{8}{23} a^{13} + \frac{68}{437} a^{12} - \frac{176}{437} a^{11} - \frac{122}{437} a^{10} - \frac{132}{437} a^{9} - \frac{189}{437} a^{8} - \frac{158}{437} a^{7} - \frac{140}{437} a^{6} - \frac{135}{437} a^{5} - \frac{18}{437} a^{4} + \frac{47}{437} a^{3} + \frac{82}{437} a^{2} - \frac{87}{437} a$, $\frac{1}{437} a^{20} + \frac{9}{437} a^{16} - \frac{9}{437} a^{15} + \frac{90}{437} a^{14} - \frac{213}{437} a^{13} + \frac{101}{437} a^{12} - \frac{212}{437} a^{11} - \frac{10}{23} a^{10} - \frac{169}{437} a^{9} + \frac{47}{437} a^{8} + \frac{1}{23} a^{7} + \frac{56}{437} a^{6} - \frac{157}{437} a^{5} + \frac{32}{437} a^{4} + \frac{3}{437} a^{3} - \frac{198}{437} a^{2} + \frac{104}{437} a - \frac{3}{19}$, $\frac{1}{8303} a^{21} + \frac{4}{8303} a^{20} - \frac{8}{8303} a^{19} + \frac{7}{8303} a^{18} + \frac{3}{8303} a^{17} + \frac{107}{8303} a^{16} + \frac{145}{8303} a^{15} - \frac{1287}{8303} a^{14} + \frac{1163}{8303} a^{13} - \frac{2765}{8303} a^{12} + \frac{2805}{8303} a^{11} - \frac{4046}{8303} a^{10} + \frac{2924}{8303} a^{9} + \frac{2357}{8303} a^{8} - \frac{2782}{8303} a^{7} - \frac{4074}{8303} a^{6} + \frac{2013}{8303} a^{5} - \frac{62}{437} a^{4} + \frac{222}{8303} a^{3} - \frac{2007}{8303} a^{2} + \frac{4058}{8303} a + \frac{33}{361}$, $\frac{1}{190969} a^{22} + \frac{1}{190969} a^{21} + \frac{170}{190969} a^{20} - \frac{7}{190969} a^{19} - \frac{37}{190969} a^{18} + \frac{79}{190969} a^{17} + \frac{1876}{190969} a^{16} + \frac{83}{190969} a^{15} + \frac{75704}{190969} a^{14} + \frac{4538}{190969} a^{13} + \frac{3231}{8303} a^{12} + \frac{50676}{190969} a^{11} - \frac{3159}{190969} a^{10} + \frac{26265}{190969} a^{9} + \frac{41903}{190969} a^{8} - \frac{76364}{190969} a^{7} - \frac{34025}{190969} a^{6} + \frac{55787}{190969} a^{5} - \frac{72985}{190969} a^{4} + \frac{41084}{190969} a^{3} + \frac{57332}{190969} a^{2} - \frac{1063}{8303} a - \frac{1}{361}$, $\frac{1}{190969} a^{23} + \frac{8}{190969} a^{21} + \frac{53}{190969} a^{20} - \frac{53}{190969} a^{19} - \frac{137}{190969} a^{18} + \frac{3}{190969} a^{17} + \frac{3704}{190969} a^{16} - \frac{601}{190969} a^{15} + \frac{69617}{190969} a^{14} - \frac{29631}{190969} a^{13} + \frac{7252}{190969} a^{12} + \frac{59164}{190969} a^{11} + \frac{84762}{190969} a^{10} + \frac{2726}{10051} a^{9} + \frac{27530}{190969} a^{8} - \frac{63001}{190969} a^{7} - \frac{55732}{190969} a^{6} - \frac{1939}{10051} a^{5} + \frac{9189}{190969} a^{4} - \frac{261}{529} a^{3} - \frac{49259}{190969} a^{2} - \frac{1716}{8303} a + \frac{36}{361}$, $\frac{1}{152761277621581954177} a^{24} + \frac{242376560013665}{152761277621581954177} a^{23} + \frac{78}{152761277621581954177} a^{22} + \frac{391326236780957}{6641794679199215399} a^{21} - \frac{137079142215798657}{152761277621581954177} a^{20} + \frac{13912791845834681}{152761277621581954177} a^{19} - \frac{14234726438863619}{152761277621581954177} a^{18} - \frac{103724187207694081}{152761277621581954177} a^{17} - \frac{1744179577865167093}{152761277621581954177} a^{16} + \frac{3091605109195568169}{152761277621581954177} a^{15} + \frac{22547551052215355720}{152761277621581954177} a^{14} + \frac{1908500295635779952}{8040067243241155483} a^{13} + \frac{71266475834510276177}{152761277621581954177} a^{12} + \frac{74427742107425630020}{152761277621581954177} a^{11} + \frac{47007527588980536717}{152761277621581954177} a^{10} + \frac{2532906094544252370}{6641794679199215399} a^{9} + \frac{11586396915082860066}{152761277621581954177} a^{8} + \frac{45042926009155343654}{152761277621581954177} a^{7} - \frac{1876312804658669051}{6641794679199215399} a^{6} + \frac{2102398571520373192}{152761277621581954177} a^{5} - \frac{60637760728327207333}{152761277621581954177} a^{4} + \frac{40278565012575859010}{152761277621581954177} a^{3} + \frac{39700932429772308102}{152761277621581954177} a^{2} - \frac{1975703641610926822}{6641794679199215399} a + \frac{123030080653691945}{288773681704313713}$, $\frac{1}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{25} + \frac{4803122791211137146775913876243114673225637736070584055822940137505222736397}{2690792818079326384527604291473579107047394412075419965376549495395503768394806695928227568979459} a^{24} + \frac{3163836574524208935230963070843843068131324945163849764772940104036519205704114796012307829906}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{23} - \frac{1594660071754268885321005971327025090779806548924517787723749396253172983235453574818349413174}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{22} + \frac{80679824655017533838495609304093901702338995562074000802763947393554389350373395508715287501247}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{21} - \frac{275323772148987817042306596499543475997622485784428463368417233653313277645329763033403579729829}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{20} - \frac{25123439220134693216725415615476012897986817244484967194962642792502402361271514353079592059627}{74917336882313876706058035273132807769898507578310376930747088582327447025308039060317493894217569} a^{19} - \frac{776343535488478138130386100502875189946267024702044531838642072278900730732675965000088943618843}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{18} - \frac{56838884299489291325202362037915454864716177469427312889588177660651394668969303151638610842258}{74917336882313876706058035273132807769898507578310376930747088582327447025308039060317493894217569} a^{17} + \frac{27852148382495566035139473216286253862914665304377931813368381042136280292212968043462178850985056}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{16} - \frac{20792406183588451470662611761951869905548633842016255242963920406591710939192581994583085118578105}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{15} + \frac{497670071786455758628673339525226862699015827524364168927059179366941340167247192613964716318977208}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{14} + \frac{25580537643893898557258031129449896852586969390309173819834065361881893697899268042581319193291123}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{13} + \frac{287516654541957611806107885790689600998296556118345437570890876850710845232253044769479119866896250}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{12} + \frac{36133986083598125612156396535307588731993737183605197154762826683308724458359368955277631108024639}{74917336882313876706058035273132807769898507578310376930747088582327447025308039060317493894217569} a^{11} - \frac{643092485251037325649998062486184127371993473259429721391290326523362090122596985374478666666348992}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{10} - \frac{470737694072741049644078638712769096152058511831476297326310781335529580086776192361596133583749308}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{9} + \frac{559360359886151273306838813449800987241659240702660924192973825957896863773112109896704615406681725}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{8} - \frac{242230694967168866548657119677596084454231042082190197602144416457399672878470328468107265792552525}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{7} - \frac{326761283835476317198064784359021335332980709745303070515480998965813250970611224996917302498233525}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{6} + \frac{492413605976712903992351938963945265405793131930097934869522286529777372252297588286823514195575442}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{5} + \frac{625349607820521178764022395293775940223373211776198715247662417932381181834255181550065895885857863}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{4} + \frac{594843308755378727111193372414937207992118352032993157182152243884004045454164634516712765010575642}{1423429400763963657415102670189523347628071643987897161684194683064221493480852742146032383990133811} a^{3} - \frac{24606030907682846915671033295761223739920903314203853170465166231499192250627722749871134647969353}{61888234815824506844134898703892319462090071477734659203660638394096586673080554006349234086527557} a^{2} - \frac{963001545459527070041799297412461497595874529716798637796497411130144209897704983082092765858279}{2690792818079326384527604291473579107047394412075419965376549495395503768394806695928227568979459} a - \frac{2445192705903302414245218746468353926806559138277223991576358136896107781989348189802347627339}{5086564873495891086063524180479355589881652952883591616968902637798683872201903016877556841171}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{13}\times C_{551629}$, which has order $7171177$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{7587167335718172362100758887201069273832791047411359313631380202282031923268}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{25} - \frac{186178386883916266946216638444509447500392653471263247141363508992806747}{63912153733261769689372866443269180369405125905441741814495874734383475861373476337} a^{24} - \frac{592097239592468417993464860047883174734442309962764426868060846915670322505199}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{23} + \frac{978983181209032864459893132303128939147086961865041045123381293002505286228549}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{22} - \frac{30388861588415274488342511549668288578921912581212833493364736713617007843947965}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{21} + \frac{56778739133409281862994821005011126423269008157480548283387513251555952200030006}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{20} - \frac{897321686059332410974408116303970222330678916842117954953095058779127121011426561}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{19} + \frac{87846981095067804718278314404872863117424854868460276041410228452534043371835395}{1779448911836604008719907702552073495548174294946246390519385143920466248982450999067} a^{18} - \frac{18726920342905278996027925753912282287583501288236137667500671328704514902920114025}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{17} + \frac{1534501270079829563032138647798716035473159629403141228069727742941787765948893519}{1779448911836604008719907702552073495548174294946246390519385143920466248982450999067} a^{16} - \frac{253498739746898110228033887113062289658334842479504294659411819084340134581215629862}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{15} + \frac{267692840610415824903330051404831616016858081166698294485589383031574674405871989330}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{14} - \frac{2428593243062991788668124001950308208279361918363530084124373679283717147093073218141}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{13} + \frac{1574935121688266323449622473284874868588077560928251463980116353616947302161897404414}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{12} - \frac{17043808159118219807054884029722867935732297631962179974654841159521131464792568710848}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{11} + \frac{3376184988131873524487175737338252009815674729458438452416027986398933146821390372257}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{10} - \frac{87593475875522141124576467190524024632817676347177641059730136234104678627652526000504}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{9} - \frac{466536234966231502273853849703331604877759945932406248692856637827529857776628968126}{1779448911836604008719907702552073495548174294946246390519385143920466248982450999067} a^{8} - \frac{328426021918654869527328719960438636266105126252150594850963830815056556491888431404643}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{7} - \frac{120232794179029121492716773030730430982958789837228777070108675551346086814733958587651}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{6} - \frac{814762768695917984567732955243366125562425745375197572661141041869698416877206224741545}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{5} - \frac{257567808517309562584605160013456218134717550090176330132967257018287785897211346710697}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{4} - \frac{1189523462492213318521689669428655765781755569169642164664703441821894914614468915155163}{33809529324895476165678246348489396415415311603978681419868317734488858730666568982273} a^{3} - \frac{17778310028004836462960277036079987411945534307814170966245318923983044787155347155494}{1469979535865020702855575928195191148496317895825160061733405118890819944811589955751} a^{2} - \frac{1912555187475365618071675816904666674871169631332457119475224780251995709327815901191}{63912153733261769689372866443269180369405125905441741814495874734383475861373476337} a - \frac{1512123122506913137446771432292015724071061034516931944294141602901398710838522}{120816925771761379374995966811472930755019141598188547853489366227568007299382753} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11194599795188.498 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{26}$ (as 26T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 13.13.542800770374370512771595361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $26$ R $26$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ R $26$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ $26$ $26$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$13$13.13.24.1$x^{13} - 13 x^{12} + 13$$13$$1$$24$$C_{13}$$[2]$
13.13.24.1$x^{13} - 13 x^{12} + 13$$13$$1$$24$$C_{13}$$[2]$